proof by induction of affine subsets

Hi, I've got a project at uni and part of it is this proof by induction which i am terrible at. I can never work out where to sub in n+1 or n+2 n-1 or whatever it may be so any help much appreciated!!

An affine subset of is a non-empty subset of with the property that whenever and

i) Let be an affine subset of . Prove by induction on that, if and with , then

(1)

belongs to .

ii) A sum of the form (1) is called an affine combination of . Prove that, given a non-empty subset of , the set consisting of all afine combinations of elements of is an affine subset of and is the smallest affine subset of containing . This set is called the affine span of .

Hi, I've got a project at uni and part of it is this proof by induction which i am terrible at. I can never work out where to sub in n+1 or n+2 n-1 or whatever it may be so any help much appreciated!!

An affine subset of is a non-empty subset of with the property that whenever and

i) Let be an affine subset of . Prove by induction on that, if and with , then

(1)

belongs to .

ii) A sum of the form (1) is called an affine combination of . Prove that, given a non-empty subset of , the set consisting of all afine combinations of elements of is an affine subset of and is the smallest affine subset of containing . This set is called the affine span of .

Two hints for the first part:

1. The base case "works" quite easily.

2. Assuming the result holds for , as you want to set for your induction step. You then have to show that the rest of the sum is of the form with . This, I believe, requires a bit of a trick, but think about it for a bit first.